Question

1. determine the y-intercept of a line that passes through (-5, 2) and is perpendicular to $y = -\frac{1}{7}x + 11$.

Explanation:

Step1: Find the slope of the perpendicular line

The slope of the given line \( y = -\frac{1}{7}x + 11 \) is \( m_1 = -\frac{1}{7} \). The slope of a line perpendicular to it, \( m_2 \), satisfies \( m_1 \times m_2 = -1 \). So, \( -\frac{1}{7} \times m_2 = -1 \), which gives \( m_2 = 7 \).

Step2: Use point - slope form to find the equation of the line

The point - slope form of a line is \( y - y_1 = m(x - x_1) \), where \( (x_1,y_1)=(-5,2) \) and \( m = 7 \). Substituting these values, we get \( y - 2 = 7(x + 5) \).

Step3: Simplify the equation to slope - intercept form

Expand the right - hand side: \( y - 2 = 7x+35 \). Then, add 2 to both sides: \( y=7x + 35+2 \), so \( y = 7x+37 \).

Step4: Identify the y - intercept

In the slope - intercept form \( y=mx + b \), the y - intercept is \( b \). For the line \( y = 7x+37 \), the y - intercept is 37.

Answer:

The y - intercept of the line is 37.